The Model

Model Overview

This model tracks susceptibles, pre-symptomatic, asymptomatic, symptomatic, diagnosed, hopspitalized, dead and recovered hosts. The following compartments are included:

  • S - uninfected and susceptible individuals
  • E - individuals who are infected and do not yet show symptoms. Those individuals can be infectious.
  • Ia - individuals who are infected and do not show symptoms. Those individuals are likely infectious, but the model allows to adjust this.
  • Isu - individuals who are infected and show symptoms. Those individuals are likely infectious. Individuals in this compartment never get diagnosed and recover.
  • Isd - individuals who are infected and show symptoms. Those individuals are likely infectious. Individuals in this compartment will get diagnosed and move to C.
  • C - individuals who have been diagnosed as cases. Those individuals are likely isolated and not infectious, but the model allows to adjust this.
  • H - individuals who have been hospitalized. Those individuals are likely isolated and not infectious, but the model allows to adjust this.
  • R - recovered/removed individuals. Those individuals have recovered and are immune.
  • D - individuals who died from the infection.

To allow more realistic distributions of movement through compartments, several of these compartments are internally split into dummy stages. Dummy stages are currently:

  • E - 4 compartments
  • Ia - 4 compartments
  • Isu - 4 compartments
  • Isd - 4 compartments
  • C - 4 compartments
  • H - 4 compartments

The included processes/mechanisms are the following:

  • Susceptible individuals (\(S\)`) can become infected by individuals in the \(E\), \(I_a\), \(I_s\), \(I_d\), \(C\), and \(H\) stages. Rates of transmission from these stages can be adjusted individually.
  • All infected hosts first enter the presymptomatic stage and move through the \(E\) dummy compartments. At the end of the \(E\) stage, a fraction moves into the \(I_a\) stage, another fraction moves into the \(I_{su}\) stage, the remainder into the \(I_{sd}\) stage.
  • Individuals in the asymptomatic \(I_a\) stage are assumed to never be diagnosed and eventually recover.
  • Individuals in the symptomatic \(I_{su}\) stage are assumed to remain undiagnosed and eventually recover.
  • Individuals in the symptomatic \(I_{sd}\) stage are assumed to be diagnosed after some time and move into the \(C\) stage.
  • A fraction of individuals in the \(C\) stage will naturally recover, without the need for hospitalization. The remainder moves into the \(H\) stage.
  • A fraction of individuals in the \(H\) stage will recover, the remainder dies.

Interventions

The following interventions are implemented:

  • Social distancing, is assumed to reduce all transmission rates by some factor. This is provided as covariate based on mobility data.
  • Increased speed of detection, which is assumed to increase at a sigmoidal manner up to a maximum value.
  • Increased level of detection, which is assumed to increase the fraction of symptomatic individuals that move into the \(I_{sd}\) compartment and will eventually be diagnosed. It is assumed to increase at a sigmoidal manner up to a maximum value.

Model Implementation

The flow diagram for the model implemented in this app is:

MIF trace plot

MIF trace plot

Results

In-Sample Trajectories

Status-quo Projections

Parameter Estimates

[1] “Alabama”

MLE Names
0.4757805 beta_s
0.9998718 max_detect_frac
2.5943251 detect_rampup
22.6599060 t_half_detect
0.1400807 frac_hosp
0.5039694 frac_dead
7.6554127 theta_cases
2.5933101 theta_hosps
3.0579666 theta_deaths
0.0405373 sigma_dw

[1] “Alaska”

MLE Names
0.2414733 beta_s
0.5689251 max_detect_frac
4.1508068 detect_rampup
20.6436619 t_half_detect
0.1258337 frac_hosp
0.2471585 frac_dead
2.4390044 theta_cases
202.8470086 theta_hosps
137.2753799 theta_deaths
0.0101502 sigma_dw

[1] “Arizona”

MLE Names
0.4952153 beta_s
0.2979032 max_detect_frac
10.8167889 detect_rampup
15.4125146 t_half_detect
0.1297715 frac_hosp
0.7980275 frac_dead
8.3666859 theta_cases
2.6056221 theta_hosps
2.0019456 theta_deaths
0.1095120 sigma_dw

[1] “Arkansas”

MLE Names
0.3947922 beta_s
0.7710313 max_detect_frac
1.8091734 detect_rampup
26.1276543 t_half_detect
0.0693480 frac_hosp
0.4900302 frac_dead
1.7732918 theta_cases
0.2584414 theta_hosps
853.7574251 theta_deaths
0.0131875 sigma_dw

[1] “California”

MLE Names
0.7683454 beta_s
0.7672789 max_detect_frac
0.5753677 detect_rampup
1.1116324 t_half_detect
0.0797148 frac_hosp
0.9951985 frac_dead
2.4105247 theta_cases
44.5863558 theta_hosps
10.0492378 theta_deaths
0.3388836 sigma_dw

[1] “Colorado”

MLE Names
0.5796489 beta_s
0.8147040 max_detect_frac
0.0625121 detect_rampup
7716.0305980 t_half_detect
0.2286327 frac_hosp
0.3959673 frac_dead
1.3471353 theta_cases
1.2006521 theta_hosps
2.6854950 theta_deaths
0.5730955 sigma_dw

[1] “Connecticut”

MLE Names
0.6423680 beta_s
0.9999891 max_detect_frac
4.1357362 detect_rampup
12.2856042 t_half_detect
0.1842007 frac_hosp
0.9997699 frac_dead
0.6352912 theta_cases
0.0136972 theta_hosps
1.9868475 theta_deaths
0.1594647 sigma_dw

[1] “Delaware”

MLE Names
0.4482800 beta_s
0.7082024 max_detect_frac
3.2860251 detect_rampup
27.2386718 t_half_detect
0.8159204 frac_hosp
0.0877455 frac_dead
1.2834403 theta_cases
0.0526792 theta_hosps
93.3603290 theta_deaths
0.0098450 sigma_dw

[1] “Florida”

MLE Names
0.6373357 beta_s
0.9968498 max_detect_frac
2.1029100 detect_rampup
16.9467417 t_half_detect
0.2079825 frac_hosp
0.3865696 frac_dead
3.5265944 theta_cases
2.5916152 theta_hosps
3.4153090 theta_deaths
0.0160608 sigma_dw

[1] “Georgia”

MLE Names
0.5678391 beta_s
0.9844980 max_detect_frac
2.9299371 detect_rampup
18.5963284 t_half_detect
0.2225743 frac_hosp
0.4927554 frac_dead
6.2849852 theta_cases
2.4615379 theta_hosps
1.2753436 theta_deaths
0.3609775 sigma_dw

[1] “Hawaii”

MLE Names
0.3966659 beta_s
0.9083725 max_detect_frac
2.8231221 detect_rampup
35.7852040 t_half_detect
0.1665278 frac_hosp
0.1905330 frac_dead
4.2991493 theta_cases
0.9179250 theta_hosps
114.8822052 theta_deaths
0.0051709 sigma_dw

[1] “Idaho”

MLE Names
3.8589378 beta_s
0.9944125 max_detect_frac
10.8917256 detect_rampup
31.6025373 t_half_detect
0.0562462 frac_hosp
0.9834307 frac_dead
0.3940830 theta_cases
24.3285162 theta_hosps
60.1337033 theta_deaths
0.0050602 sigma_dw

[1] “Illinois”

MLE Names
0.6517297 beta_s
0.9980447 max_detect_frac
2.7793398 detect_rampup
19.0269305 t_half_detect
0.1057170 frac_hosp
0.9966830 frac_dead
14.9430351 theta_cases
0.0456789 theta_hosps
3.7933688 theta_deaths
0.3862127 sigma_dw

[1] “Indiana”

MLE Names
0.4893433 beta_s
0.9988267 max_detect_frac
4.3619277 detect_rampup
20.6141564 t_half_detect
0.1559074 frac_hosp
0.9753322 frac_dead
42.7062459 theta_cases
0.0021651 theta_hosps
2.1782254 theta_deaths
0.5666347 sigma_dw

[1] “Iowa”

MLE Names
0.5140032 beta_s
0.9819411 max_detect_frac
3.1574252 detect_rampup
27.6792387 t_half_detect
0.2631907 frac_hosp
0.2227324 frac_dead
8.5172759 theta_cases
2.5045087 theta_hosps
17.1260586 theta_deaths
0.5526685 sigma_dw

[1] “Kansas”

MLE Names
0.4670485 beta_s
0.6249303 max_detect_frac
8.2538338 detect_rampup
13.4507017 t_half_detect
0.0928718 frac_hosp
0.2931695 frac_dead
0.7572510 theta_cases
1.3819168 theta_hosps
14.9143112 theta_deaths
0.0011814 sigma_dw

[1] “Kentucky”

MLE Names
0.4504021 beta_s
0.5968999 max_detect_frac
17.2383267 detect_rampup
12.7250076 t_half_detect
0.3211911 frac_hosp
0.2042541 frac_dead
1.3547212 theta_cases
0.5002202 theta_hosps
8.4496537 theta_deaths
0.0075724 sigma_dw

[1] “Louisiana”

MLE Names
0.7563637 beta_s
0.9808402 max_detect_frac
5.4206545 detect_rampup
1.6995865 t_half_detect
0.9995250 frac_hosp
0.1277534 frac_dead
3.0558676 theta_cases
1.4097063 theta_hosps
2.9749771 theta_deaths
0.6961753 sigma_dw

[1] “Maine”

MLE Names
0.4015399 beta_s
0.2885138 max_detect_frac
3.7710738 detect_rampup
14.7487996 t_half_detect
0.1229750 frac_hosp
0.4926108 frac_dead
5.9632682 theta_cases
6148.9441984 theta_hosps
260.9166447 theta_deaths
0.0000794 sigma_dw

[1] “Maryland”

MLE Names
0.6033276 beta_s
0.9960984 max_detect_frac
3.1837610 detect_rampup
23.3125629 t_half_detect
0.2348962 frac_hosp
0.4195236 frac_dead
6.8681600 theta_cases
6.9297887 theta_hosps
7.2241614 theta_deaths
0.0231141 sigma_dw

[1] “Massachusetts”

MLE Names
0.7729154 beta_s
0.9154106 max_detect_frac
1.9961991 detect_rampup
0.5285161 t_half_detect
0.1939738 frac_hosp
0.9831738 frac_dead
0.7305065 theta_cases
1.0836458 theta_hosps
1.7022323 theta_deaths
0.2079888 sigma_dw

[1] “Michigan”

MLE Names
0.6681018 beta_s
0.9993098 max_detect_frac
2.1230627 detect_rampup
0.5034660 t_half_detect
0.1630305 frac_hosp
0.7297034 frac_dead
6.2854481 theta_cases
11.8635303 theta_hosps
2.1390859 theta_deaths
0.8461073 sigma_dw

[1] “Minnesota”

MLE Names
0.5054076 beta_s
0.6359424 max_detect_frac
1.5800931 detect_rampup
31.9776450 t_half_detect
0.2300679 frac_hosp
0.6281031 frac_dead
6.7543069 theta_cases
6.9099566 theta_hosps
13.5383492 theta_deaths
0.0000234 sigma_dw

[1] “Mississippi”

MLE Names
2.5886407 beta_s
0.9653309 max_detect_frac
5.9199069 detect_rampup
42.8752835 t_half_detect
0.3523971 frac_hosp
0.0452218 frac_dead
0.1670806 theta_cases
0.3147402 theta_hosps
1.0696476 theta_deaths
0.0308805 sigma_dw

[1] “Missouri”

MLE Names
0.5056369 beta_s
0.9965313 max_detect_frac
4.2806389 detect_rampup
18.4924466 t_half_detect
0.0893818 frac_hosp
0.9992059 frac_dead
2.6695570 theta_cases
1.9539823 theta_hosps
2.2036258 theta_deaths
0.0000027 sigma_dw

[1] “Montana”

MLE Names
0.2190737 beta_s
0.9819180 max_detect_frac
5.5007664 detect_rampup
20.1344872 t_half_detect
0.1329944 frac_hosp
0.2975285 frac_dead
8.5093143 theta_cases
29.9828071 theta_hosps
9.1483486 theta_deaths
0.0624302 sigma_dw

[1] “Nebraska”

MLE Names
0.7026193 beta_s
0.0171538 max_detect_frac
0.0541000 detect_rampup
37.0928299 t_half_detect
0.9973451 frac_hosp
0.0322449 frac_dead
3.0009511 theta_cases
0.0001652 theta_hosps
4.5408271 theta_deaths
0.0018579 sigma_dw

[1] “Nevada”

MLE Names
0.4914973 beta_s
0.7520749 max_detect_frac
3.5079474 detect_rampup
19.9241852 t_half_detect
0.6409380 frac_hosp
0.1100404 frac_dead
6.5147477 theta_cases
967.8070642 theta_hosps
7.4327163 theta_deaths
0.0016814 sigma_dw

[1] “New Hampshire”

MLE Names
0.4553380 beta_s
0.3978741 max_detect_frac
3.4541220 detect_rampup
20.2656428 t_half_detect
0.1238897 frac_hosp
0.6179200 frac_dead
0.6943660 theta_cases
1.4791546 theta_hosps
1.1575968 theta_deaths
0.0317890 sigma_dw

[1] “New Jersey”

MLE Names
0.7531299 beta_s
0.9890144 max_detect_frac
2.9788748 detect_rampup
13.1645477 t_half_detect
0.1853898 frac_hosp
0.7050469 frac_dead
11.1884673 theta_cases
13.2274055 theta_hosps
2.1934997 theta_deaths
1.0887049 sigma_dw

[1] “New Mexico”

MLE Names
4.357194e-01 beta_s
9.976292e-01 max_detect_frac
2.533738e+00 detect_rampup
4.710459e+01 t_half_detect
2.015578e-01 frac_hosp
4.145089e-01 frac_dead
3.893231e+00 theta_cases
2.050290e-01 theta_hosps
6.868251e+04 theta_deaths
4.011400e-03 sigma_dw

[1] “New York”

MLE Names
0.8069368 beta_s
0.9631430 max_detect_frac
0.9420071 detect_rampup
1.6921558 t_half_detect
0.2205142 frac_hosp
0.3037073 frac_dead
1.1571367 theta_cases
2.5602950 theta_hosps
4.6052996 theta_deaths
1.1453825 sigma_dw

[1] “North Carolina”

MLE Names
0.4912003 beta_s
0.7130152 max_detect_frac
3.4637861 detect_rampup
21.6184326 t_half_detect
0.9676310 frac_hosp
0.0724228 frac_dead
7.9707225 theta_cases
4773.6180369 theta_hosps
3.5601530 theta_deaths
0.0005844 sigma_dw

[1] “North Dakota”

MLE Names
0.3604785 beta_s
0.9460589 max_detect_frac
2.3013497 detect_rampup
44.0761575 t_half_detect
0.0936381 frac_hosp
0.4513159 frac_dead
2.4972940 theta_cases
9547.0722228 theta_hosps
161.7090704 theta_deaths
0.0051387 sigma_dw

[1] “Ohio”

MLE Names
0.4773143 beta_s
0.9485901 max_detect_frac
4.2277360 detect_rampup
17.3502993 t_half_detect
0.1827068 frac_hosp
0.4313134 frac_dead
6.9363940 theta_cases
10.2694614 theta_hosps
4.4857994 theta_deaths
0.8387246 sigma_dw

[1] “Oklahoma”

MLE Names
0.4203167 beta_s
0.7566808 max_detect_frac
4.1509598 detect_rampup
16.8617860 t_half_detect
0.2006356 frac_hosp
0.4310370 frac_dead
4.2912841 theta_cases
0.6320082 theta_hosps
1.7204058 theta_deaths
0.0038317 sigma_dw

[1] “Oregon”

MLE Names
0.4327268 beta_s
0.5119361 max_detect_frac
4.1487139 detect_rampup
16.7735740 t_half_detect
0.2107754 frac_hosp
0.2229556 frac_dead
1.5798511 theta_cases
1.4868724 theta_hosps
16.5543752 theta_deaths
0.0000216 sigma_dw

[1] “Pennsylvania”

MLE Names
0.6702931 beta_s
0.9079422 max_detect_frac
2.9009401 detect_rampup
19.3592778 t_half_detect
0.1681187 frac_hosp
0.9983662 frac_dead
62.7174540 theta_cases
2.4521287 theta_hosps
0.9980466 theta_deaths
1.0248704 sigma_dw

[1] “Rhode Island”

MLE Names
0.8198189 beta_s
0.1017122 max_detect_frac
0.8074645 detect_rampup
284.9928463 t_half_detect
0.1392710 frac_hosp
0.5527588 frac_dead
38.4599160 theta_cases
0.7603183 theta_hosps
1.7041735 theta_deaths
0.4946123 sigma_dw

[1] “South Carolina”

MLE Names
0.4294693 beta_s
0.9901551 max_detect_frac
1.5790809 detect_rampup
28.5578336 t_half_detect
0.1998226 frac_hosp
0.4088911 frac_dead
0.3890405 theta_cases
0.0453772 theta_hosps
0.6817332 theta_deaths
0.0039764 sigma_dw

[1] “South Dakota”

MLE Names
0.8703450 beta_s
0.0067903 max_detect_frac
8.2736680 detect_rampup
12.5196208 t_half_detect
0.0916336 frac_hosp
0.2075713 frac_dead
5.4617063 theta_cases
1885.5555854 theta_hosps
2.4653109 theta_deaths
0.8479423 sigma_dw

[1] “Tennessee”

MLE Names
0.4725450 beta_s
0.9908922 max_detect_frac
2.1543169 detect_rampup
21.4147435 t_half_detect
0.0891754 frac_hosp
0.3192161 frac_dead
5.2637242 theta_cases
2.1406402 theta_hosps
2.1739394 theta_deaths
0.7909954 sigma_dw

[1] “Texas”

MLE Names
0.6012227 beta_s
0.8440029 max_detect_frac
5.1786280 detect_rampup
18.6295301 t_half_detect
0.0528255 frac_hosp
0.9986054 frac_dead
4.5271595 theta_cases
353.3343351 theta_hosps
4.7091918 theta_deaths
0.2234753 sigma_dw

[1] “Utah”

MLE Names
0.4631379 beta_s
0.9993058 max_detect_frac
2.4030697 detect_rampup
22.3376673 t_half_detect
0.0832513 frac_hosp
0.2127126 frac_dead
13.2697297 theta_cases
3726.6847360 theta_hosps
2443.8913546 theta_deaths
0.2688527 sigma_dw

[1] “Vermont”

MLE Names
2.8242303 beta_s
0.8967214 max_detect_frac
9.6074520 detect_rampup
49.9365148 t_half_detect
0.1233905 frac_hosp
0.2797252 frac_dead
0.8904855 theta_cases
0.1178884 theta_hosps
1782.0967756 theta_deaths
0.0002420 sigma_dw

[1] “Virginia”

MLE Names
0.5299923 beta_s
0.9242610 max_detect_frac
2.7323813 detect_rampup
28.0106273 t_half_detect
0.2304851 frac_hosp
0.3778277 frac_dead
4.2367129 theta_cases
0.8712811 theta_hosps
2.5009447 theta_deaths
0.2249052 sigma_dw

[1] “Washington”

MLE Names
0.5110536 beta_s
0.0075791 max_detect_frac
0.2123196 detect_rampup
0.1560973 t_half_detect
0.8123395 frac_hosp
0.1284921 frac_dead
0.4268339 theta_cases
22.4491855 theta_hosps
7.2360925 theta_deaths
0.0023018 sigma_dw

[1] “West Virginia”

MLE Names
0.3280613 beta_s
0.5807087 max_detect_frac
5.3376010 detect_rampup
23.5649314 t_half_detect
0.5489887 frac_hosp
0.1161151 frac_dead
3.2819078 theta_cases
0.0000103 theta_hosps
38.1040870 theta_deaths
0.0108008 sigma_dw

[1] “Wisconsin”

MLE Names
0.5019510 beta_s
0.7510068 max_detect_frac
3.3567859 detect_rampup
16.5550132 t_half_detect
0.1682813 frac_hosp
0.3589507 frac_dead
6.4120375 theta_cases
5.3823749 theta_hosps
2.7594721 theta_deaths
0.0178186 sigma_dw

[1] “Wyoming”

MLE Names
0.2944637 beta_s
0.2163054 max_detect_frac
8.5549068 detect_rampup
15.0829918 t_half_detect
0.1164160 frac_hosp
0.1397636 frac_dead
0.9861360 theta_cases
3.3417054 theta_hosps
12.1024765 theta_deaths
0.0521916 sigma_dw